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180 Relations Example 11.8 Prove the following proposition. Proposition Let n ∈ N. The relation ≡ (mod n) on the set Z is reflexive, symmetric and transitive. Proof. First we will show that ≡ (mod n) is reflexive. Take any integer x ∈ Z, and observe that n|0, so n | (x − x). By definition of congruence modulo n, we have x ≡ x (mod n). This shows x ≡ x (mod n) for every x ∈ Z, so ≡ (mod n) is reflexive. Next, we will show that ≡ (mod n) is symmetric. For this, we must show that for all x, y ∈ Z, the condition x ≡ y (mod n) implies that y ≡ x (mod n). We use direct proof. Suppose x ≡ y (mod n). Thus n | (x − y) by definition of congruence modulo n. Then x − y = na for some a ∈ Z by definition of divisibility. Multiplying both sides by −1 gives y − x = n(−a). Therefore n | (y − x), and this means y ≡ x (mod n). We’ve shown that x ≡ y (mod n) implies that y ≡ x (mod n), and this means ≡ (mod n) is symmetric. Finally we will show that ≡ (mod n) is transitive. For this we must show that if x ≡ y (mod n) and y ≡ z (mod n), then x ≡ z (mod n). Again we use direct proof. Suppose x ≡ y (mod n) and y ≡ z (mod n). This means n | (x − y) and n | (y − z). Therefore there are integers a and b for which x − y = na and y − z = nb. Adding these two equations, we obtain x−z = na+nb. Consequently, x−z = n(a+b), so n | (x−z), hence x ≡ z (mod n). This completes the proof that ≡ (mod n) is transitive. The past three paragraphs have shown that ≡ (mod n) is reflexive, symmetric and transitive, so the proof is complete. ■ As you continue your mathematical education, you will find that the reflexive, symmetric and transitive properties take on special significance in a variety of settings. In preparation for this, the next section explores further consequences of these relations. But first you should work some of the following exercises. Exercises for Section 11.1 1. Consider the relation R = { (a, a),(b, b),(c, c),(d, d),(a, b),(b, a) } on set A = { a, b, c, d } . Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. 2. Consider the relation R = { (a, b),(a, c),(c, c),(b, b),(c, b),(b, c) } on the set A = { a, b, c } . Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. 3. Consider the relation R = { (a, b),(a, c),(c, b),(b, c) } on the set A = { a, b, c } . Is R reflexive? Symmetric? Transitive? If a property does not hold, say why.
Properties of Relations 181 4. Let A = { a, b, c, d } . Suppose R is the relation R = { (a, a),(b, b),(c, c),(d, d),(a, b),(b, a),(a, c),(c, a), (a, d),(d, a),(b, c),(c, b),(b, d),(d, b),(c, d),(d, c) } . Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. 5. Consider the relation R = { (0,0),( 2,0),(0, 2),( 2, 2) } on R. Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. 6. Consider the relation R = { (x, x) : x ∈ Z } on Z. Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. What familiar relation is this? 7. There are 16 possible different relations R on the set A = { a, b } . Describe all of them. (A picture for each one will suffice, but don’t forget to label the nodes.) Which ones are reflexive? Symmetric? Transitive? 8. Define a relation on Z as xR y if |x−y| < 1. Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. What familiar relation is this? 9. Define a relation on Z by declaring xR y if and only if x and y have the same parity. Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. What familiar relation is this? 10. Suppose A ≠ . Since ⊆ A × A, the set R = is a relation on A. Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. 11. Suppose A = { a, b, c, d } and R = { (a, a),(b, b),(c, c),(d, d) } . Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. 12. Prove that the relation | (divides) on the set Z is reflexive and transitive. (Use Example 11.8 as a guide if you are unsure of how to proceed.) 13. Consider the relation R = { (x, y) ∈ R × R : x − y ∈ Z } on R. Prove that this relation is reflexive, symmetric and transitive. 14. Suppose R is a symmetric and transitive relation on a set A, and there is an element a ∈ A for which aRx for every x ∈ A. Prove that R is reflexive. 15. Prove or disprove: If a relation is symmetric and transitive, then it is also reflexive. 16. Define a relation R on Z by declaring that xR y if and only if x 2 ≡ y 2 (mod 4). Prove that R is reflexive, symmetric and transitive. 17. Modifying the above Exercise 8 (above) slightly, define a relation ∼ on Z as x ∼ y if and only if |x− y| ≤ 1. Say whether ∼ is reflexive. Is it symmetric? Transitive? 18. The table on page 177 shows that relations on Z may obey various combinations of the reflexive, symmetric and transitive properties. In all, there are 2 3 = 8 possible combinations, and the table shows 5 of them. (There is some redundancy, as ≤ and | have the same type.) Complete the table by finding examples of relations on Z for the three missing combinations.
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Book of Proof Richard Hammack Virgi
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To my students
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v II How to Prove Conditional State
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Preface In writing this book I have
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ix The book is organized into four
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Part I Fundamentals
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4 Sets Some sets are so significant
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6 Sets These last three examples hi
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8 Sets 1.2 The Cartesian Product Gi
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10 Sets We can also take Cartesian
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12 Sets This idea of “making” a
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14 Sets This is a subset C ⊆ R 2
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16 Sets y y y x x x (a) (b) (c) Fig
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18 Sets B A ∪ B A ∩ B A − B A
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20 Sets Example 1.7 If P is the set
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22 Sets We can also think of A ∩
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24 Sets 1.8 Indexed Sets When a mat
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26 Sets Example 1.11 Here the sets
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28 Sets 1.9 Sets that Are Number Sy
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30 Sets Russell’s paradox arises
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32 Logic In proving theorems, we ap
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34 Logic R(f , g) : The function f
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36 Logic 2.2 And, Or, Not The word
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38 Logic To conclude this section,
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40 Logic You can think of P ⇒ Q a
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42 Logic that it’s impossible tha
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44 Logic P if and only if Q. P is a
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46 Logic Notice that when we plug i
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48 Logic There are two pairs of log
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50 Logic Likewise, a statement such
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52 Logic 2.8 More on Conditional St
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54 Logic Example 2.9 Consider Goldb
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56 Logic Example 2.10 Consider nega
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58 Logic Example 2.13 Negate the fo
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60 Logic 2.12 An Important Note It
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62 Counting Occasionally we may get
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64 Counting To answer this question
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66 Counting we can choose any one o
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68 Counting 3.2 Factorials In worki
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70 Counting We close this section w
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72 Counting Definition 3.2 If n and
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74 Counting Fact 3.3 ( ( ) n n! n I
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76 Counting 3.4 Pascal’s Triangle
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78 Counting coefficients 1 2 1. Sim
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80 Counting We seek the number of 3
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82 Counting 5. How many 7-digit bin
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CHAPTER 4 Direct Proof It is time t
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Definitions 87 4.2 Definitions A pr
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Definitions 89 it necessary to defi
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Direct Proof 91 So the setup for di
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Direct Proof 93 Here is another exa
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Direct Proof 95 This proposition te
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Treating Similar Cases 97 Propositi
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Treating Similar Cases 99 19. Suppo
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Contrapositive Proof 101 Since P
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Congruence of Integers 103 Proposit
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Mathematical Writing 105 Propositio
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Mathematical Writing 107 Since a |
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CHAPTER 6 Proof by Contradiction We
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Proving Statements with Contradicti
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Proving Conditional Statements by C
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Some Words of Advice 115 for intege
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Part III More on Proof
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120 Proving Non-Conditional Stateme
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Page 140 and 141: 130 Proofs Involving Sets Generally
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Page 146 and 147: 136 Proofs Involving Sets Thus, sin
Page 148 and 149: 138 Proofs Involving Sets Though it
Page 150 and 151: 140 Proofs Involving Sets If we add
Page 152 and 153: 142 Proofs Involving Sets so that d
Page 154 and 155: CHAPTER 9 Disproof E ver since Chap
Page 156 and 157: 146 Disproof deciding whether a sta
Page 158 and 159: 148 Disproof Example 9.2 Conjecture
Page 160 and 161: 150 Disproof 9.3 Disproof by Contra
Page 162 and 163: CHAPTER 10 Mathematical Induction T
Page 164 and 165: 154 Mathematical Induction This pic
Page 166 and 167: 156 Mathematical Induction To round
Page 168 and 169: 158 Mathematical Induction Proposit
Page 170 and 171: 160 Mathematical Induction Strong i
Page 174 and 175: 164 Mathematical Induction We next
Page 178 and 179: 168 Mathematical Induction 13. For
Page 181: Part IV Relations, Functions and Ca
Page 184 and 185: 174 Relations The set R encodes the
Page 186 and 187: 176 Relations Exercises for Section
Page 188 and 189: 178 Relations Example 11.7 Here A =
Page 192 and 193: 182 Relations 11.2 Equivalence Rela
Page 194 and 195: 184 Relations We close this section
Page 196 and 197: 186 Relations 11.3 Equivalence Clas
Page 198 and 199: 188 Relations Notationally, the uni
Page 200 and 201: 190 Relations In a similar vein, [2
Page 202 and 203: 192 Relations Exercises for Section
Page 204 and 205: CHAPTER 12 Functions You know from
Page 206 and 207: 196 Functions value a to the output
Page 208 and 209: 198 Functions To answer this, first
Page 210 and 211: 200 Functions For more concrete exa
Page 212 and 213: 202 Functions To see that g is surj
Page 214 and 215: 204 Functions Although the underlyi
Page 216 and 217: 206 Functions 12.4 Composition You
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Page 224 and 225: 214 Functions If you continue your
Page 226 and 227: 216 Cardinality of Sets On the othe
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Page 230 and 231: 220 Cardinality of Sets And saying
Page 232 and 233: 222 Cardinality of Sets Here is a s
Page 234 and 235: 224 Cardinality of Sets Beginning a
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Page 238 and 239: 228 Cardinality of Sets f : A → P
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230 Cardinality of Sets 13.4 The Ca
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232 Cardinality of Sets and the whi
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234 Cardinality of Sets Here are so
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236 Cardinality of Sets On the face
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Solutions Chapter 1 Exercises Secti
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240 Solutions Sketch the following
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242 Solutions 5. Sketch the sets X
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244 Solutions Section 1.8 1. Suppos
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246 Solutions Section 2.4 Without c
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248 Solutions 7. P ⇒ Q = (P∧
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250 Solutions Section 2.10 Negate t
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252 Solutions (b) How many such cod
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254 Solutions 5. How many 7-digit b
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256 Solutions 15. If n ∈ Z, then
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258 Solutions 7. Proposition Suppos
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260 Solutions 27. If a ≡ 0 (mod 4
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262 Solutions 7. If a, b ∈ Z, the
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264 Solutions This expresses a 3 +
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266 Solutions 19. If n ∈ N, then
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268 Solutions Chapter 8 Exercises 1
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270 Solutions 19. Prove that {9 n :
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272 Solutions 3. If n ∈ Z and n 5
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274 Solutions 25. For all a, b, c
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276 Solutions 5. If n ∈ N, then 2
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278 Solutions (2) Now assume the st
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280 Solutions 23. Use induction to
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282 Solutions Thus, with n + 1 line
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284 Solutions a b a b a b a b a b a
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286 Solutions Therefore 3x − 5z i
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288 Solutions Chapter 12 Exercises
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290 Solutions 13. Consider the func
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292 Solutions Section 12.5 Exercise
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294 Solutions 13. Let f : A → B b
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296 Solutions 11. Partition N into
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298 Solutions 7. Prove or disprove:
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300 Index notation, 197 one-to-one,
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